Breaking \(e^n\) barrier for deterministic poly-time approximation of the permanent and settling Friedland's conjecture on the Monomer-Dimer Entropy

Two important breakthroughs on the permanent had been accomplished in 1998: A. Schrijver proved Schrijver-Valiant Conjecture on the minimal exponential growth of the number of perfect matchings in k-regular bipartite graphs with multiple edges; N. Linial, A. Samorodnitsky and A. Wigderson introduced a strongly poly-time deterministic algorithm to approximate the permanent of general non-negative matrices within the multiplicative factor en. Many things happened since them, notably the prize-winning Jerrum, Vigoda, Sinclair FPRAS for the permanent. Schrijver's lower bound was vastly generalized and improved; moreover the new proofs, based on hyperbolic(aka stable) polynomials, are transparent, easy, non-computational. Yet, until now there were no deterministic poly- time algorithms to approximate the permanent of general non-negative matrices within the multiplicative factor \(F^n\), \(F